Table of contents
CSPICE_DVSEP calculates the time derivative of the separation angle
between states.
Given:
s1,
s2 respectively, the state vector of the first and second target
bodies as seen from the observer.
help, s1
DOUBLE = Array[6]
help, s2
DOUBLE = Array[6]
An implicit assumption exists that both states lie in the
same reference frame with the same observer for the same
epoch. If this is not the case, the numerical result has
no meaning.
the call:
dvsep = cspice_dvsep( s1, s2 )
returns:
dvsep the double precision value of the time derivative of the angular
separation between `s1' and `s2'.
help, dvsep
DOUBLE = Scalar
None.
Any numerical results shown for this example may differ between
platforms as the results depend on the SPICE kernels used as input
and the machine specific arithmetic implementation.
1) Calculate the time derivative of the angular separation of
the Earth and Moon as seen from the Sun.
Use the meta-kernel shown below to load the required SPICE
kernels.
KPL/MK
File name: dvsep_ex1.tm
This meta-kernel is intended to support operation of SPICE
example programs. The kernels shown here should not be
assumed to contain adequate or correct versions of data
required by SPICE-based user applications.
In order for an application to use this meta-kernel, the
kernels referenced here must be present in the user's
current working directory.
The names and contents of the kernels referenced
by this meta-kernel are as follows:
File name Contents
--------- --------
de421.bsp Planetary ephemeris
naif0012.tls Leapseconds
\begindata
KERNELS_TO_LOAD = ( 'de421.bsp',
'naif0012.tls' )
\begintext
End of meta-kernel
Example code begins here.
PRO dvsep_ex1
;;
;; Load kernels.
;;
cspice_furnsh, 'dvsep_ex1.tm'
;;
;; An arbitrary time.
;;
BEGSTR = 'JAN 1 2009'
cspice_str2et, BEGSTR, et
;;
;; Calculate the state vectors sun to Moon, sun to earth at ET.
;;
cspice_spkezr, 'EARTH', et, 'J2000', 'NONE', 'SUN', statee, ltime
cspice_spkezr, 'MOON', et, 'J2000', 'NONE', 'SUN', statem, ltime
;;
;; Calculate the time derivative of the angular separation of
;; the earth and Moon as seen from the sun at ET.
;;
dsept = cspice_dvsep( statee, statem )
print, 'Time derivative of angular separation, rads/sec: ', dsept
;;
;; It's always good form to unload kernels after use,
;; particularly in IDL due to data persistence.
;;
cspice_kclear
END
When this program was executed on a Mac/Intel/IDL8.x/64-bit
platform, the output was:
Time derivative of angular separation, rads/sec: 3.8121194e-09
In this discussion, the notation
< V1, V2 >
indicates the dot product of vectors V1 and V2. The notation
V1 x V2
indicates the cross product of vectors V1 and V2.
To start out, note that we need consider only unit vectors,
since the angular separation of any two non-zero vectors
equals the angular separation of the corresponding unit vectors.
Call these vectors U1 and U2; let their velocities be V1 and V2.
For unit vectors having angular separation
THETA
the identity
|| U1 x U1 || = ||U1|| * ||U2|| * sin(THETA) (1)
reduces to
|| U1 x U2 || = sin(THETA) (2)
and the identity
| < U1, U2 > | = || U1 || * || U2 || * cos(THETA) (3)
reduces to
| < U1, U2 > | = cos(THETA) (4)
Since THETA is an angular separation, THETA is in the range
0 : Pi
Then letting s be +1 if cos(THETA) > 0 and -1 if cos(THETA) < 0,
we have for any value of THETA other than 0 or Pi
2 1/2
cos(THETA) = s * ( 1 - sin (THETA) ) (5)
or
2 1/2
< U1, U2 > = s * ( 1 - sin (THETA) ) (6)
At this point, for any value of THETA other than 0 or Pi,
we can differentiate both sides with respect to time (T)
to obtain
2 -1/2
< U1, V2 > + < V1, U2 > = s * (1/2)(1 - sin (THETA))
* (-2) sin(THETA)*cos(THETA)
* d(THETA)/dT (7a)
Using equation (5), and noting that s = 1/s, we can cancel
the cosine terms on the right hand side
-1
< U1, V2 > + < V1, U2 > = (1/2)(cos(THETA))
* (-2) sin(THETA)*cos(THETA)
* d(THETA)/dT (7b)
With (7b) reducing to
< U1, V2 > + < V1, U2 > = - sin(THETA) * d(THETA)/dT (8)
Using equation (2) and switching sides, we obtain
|| U1 x U2 || * d(THETA)/dT = - < U1, V2 > - < V1, U2 > (9)
or, provided U1 and U2 are linearly independent,
d(THETA)/dT = ( - < U1, V2 > - < V1, U2 > ) / ||U1 x U2|| (10)
Note for times when U1 and U2 have angular separation 0 or Pi
radians, the derivative of angular separation with respect to
time doesn't exist. (Consider the graph of angular separation
with respect to time; typically the graph is roughly v-shaped at
the singular points.)
1) If numeric overflow and underflow cases are detected, an error
is signaled by a routine in the call tree of this routine.
2) Linear dependent position components of `s1' and `s1' constitutes
a non-error exception. The function returns 0 for this case.
3) If any of the input arguments, `s1' or `s2', is undefined, an
error is signaled by the IDL error handling system.
4) If any of the input arguments, `s1' or `s2', is not of the
expected type, or it does not have the expected dimensions and
size, an error is signaled by the Icy interface.
None.
None.
ICY.REQ
None.
J. Diaz del Rio (ODC Space)
E.D. Wright (JPL)
-Icy Version 1.0.1, 31-MAY-2021 (JDR)
Edited the header to comply with NAIF standard. Added example's
problem statement and meta-kernel.
Added -Parameters, -Exceptions, -Files, -Restrictions,
-Literature_References and -Author_and_Institution sections.
Removed reference to the routine's corresponding CSPICE header from
-Abstract section.
Added arguments' type and size information in the -I/O section.
-Icy Version 1.0.0, 07-DEC-2009 (EDW)
time derivative of angular separation
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